In mathematics, Riemann's differential equation, named after Bernhard Riemann, is a generalization of the hypergeometric differential equation, allowing the regular singular points to occur anywhere on the Riemann sphere, rather than merely at 0, 1, and ∞ {\displaystyle \infty } . The equation is also known as the Papperitz equation. The hypergeometric differential equation is a second-order linear differential equation which has three regular singular points, 0, 1 and ∞ {\displaystyle \infty } . That equation admits two linearly independent solutions; near a singularity z s {\displaystyle z_{s}} , the solutions take the form x s f ( x ) {\displaystyle x^{s}f(x)} , where x = z − z s {\displaystyle x=z-z_{s}} is a local variable, and f {\displaystyle f} is locally holomorphic with f ( 0 ) ≠ 0 {\displaystyle f(0)\neq 0} . The real number s {\displaystyle s} is called the exponent of the solution at z s {\displaystyle z_{s}} . Let α, β and γ be the exponents of one solution at 0, 1 and ∞ {\displaystyle \infty } respectively; and let α′, β′ and γ′ be those of the other. Then
α + α ′ + β + β ′ + γ + γ ′ = 1. {\displaystyle \alpha +\alpha '+\beta +\beta '+\gamma +\gamma '=1.}
By applying suitable changes of variable, it is possible to transform the hypergeometric equation: Applying Möbius transformations will adjust the positions of the regular singular points, while other transformations (see below) can change the exponents at the regular singular points, subject to the exponents adding up to 1.
Definition The differential equation is given by
d 2 w d z 2 + [ 1 − α − α ′ z − a + 1 − β − β ′ z − b + 1 − γ − γ ′ z − c ] d w d z {\displaystyle {\frac {d^{2}w}{dz^{2}}}+\left[{\frac {1-\alpha -\alpha '}{z-a}}+{\frac {1-\beta -\beta '}{z-b}}+{\frac {1-\gamma -\gamma '}{z-c}}\right]{\frac {dw}{dz}}}
+ [ α α ′ ( a − b ) ( a − c ) z − a + β β ′ ( b − c ) ( b − a ) z − b + γ γ ′ ( c − a ) ( c − b ) z − c ] w ( z − a ) ( z − b ) ( z − c ) = 0. {\displaystyle +\left[{\frac {\alpha \alpha '(a-b)(a-c)}{z-a}}+{\frac {\beta \beta '(b-c)(b-a)}{z-b}}+{\frac {\gamma \gamma '(c-a)(c-b)}{z-c}}\right]{\frac {w}{(z-a)(z-b)(z-c)}}=0.}
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